Question: Simplify and expand the following expression: $ \dfrac{3n + 2}{3n - 2}-\dfrac{5n}{n + 5} $
Answer: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(3n - 2)(n + 5)$ Multiply the first term by $\dfrac{n + 5}{n + 5}$ $ \begin{align*} \dfrac{3n + 2}{3n - 2} \times \dfrac{n + 5}{n + 5} & = \dfrac{(3n + 2)(n + 5)}{(3n - 2)(n + 5)} \\ & = \dfrac{3n^2 + 17n + 10}{(3n - 2)(n + 5)}\end{align*} $ Multiply the second term by $\dfrac{3n - 2}{3n - 2}$ $ \begin{align*} \dfrac{5n}{n + 5} \times \dfrac{3n - 2}{3n - 2} & = \dfrac{(5n)(3n - 2)}{(n + 5)(3n - 2)} \\ & = \dfrac{15n^2 - 10n}{(n + 5)(3n - 2)}\end{align*} $ Now we have: $ = \dfrac{3n^2 + 17n + 10}{(3n - 2)(n + 5)} - \dfrac{15n^2 - 10n}{(n + 5)(3n - 2)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{3n^2 + 17n + 10 - (15n^2 - 10n)}{(3n - 2)(n + 5)} $ $ = \dfrac{3n^2 + 17n + 10 - 15n^2 + 10n}{(3n - 2)(n + 5)} $ $ = \dfrac{-12n^2 + 27n + 10}{(3n - 2)(n + 5)}$ Expand the denominator: $ = \dfrac{-12n^2 + 27n + 10}{3n^2 + 13n - 10}$